Finite Group Databasebeta

← #416663

Subgroups — #416663

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21172C211yesmaximal, characteristic
3#20765C211yesmaximal, characteristic
4#14354C211yesmaximal, characteristic
5#11340C212yesmaximal
6#11339C212yesmaximal
7#11211C212yesmaximal
8#11210C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10840C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#428C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2859—22no—
17#2647—22no—
18#2571—22no—
19#2552—22no—
20#2442—24no—
21#2317C2212yes—
22#2316C2212yes—
23#2285C2211yescharacteristic
24C2×#384C2212yes—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2212yes—
28C4×#117C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1949—22no—
31#1949—22no—
32#1836—22no—
33#1835—22no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36#1736C2212yes—
37#1736C2212yes—
38#1736C2211yescharacteristic
39#1736C2211yescharacteristic
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44#1444—24no—
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C2×#125—24no—
49C2×#117C2311yescharacteristic
50C2×#117C2312yes—
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—24no—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59#428—42no—
60#428—42no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#106—21no—
64C2×#106—21no—
65#423—22no—
66#423Q814yes—
67C2×C4⋊C8—22no—
68C4×M16—22no—
69C4×M16Q814yes—
70C2×#97—21no—
71C2×C4×C8—21no—
72#400—28no—
73#398—28no—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—24no—
78#389—24no—
79#385—24no—
80#384—22no—
81#384—22no—
82#384—21no—
83#384—21no—
84#384—24no—
85#384—42no—
86#384D412yes—
87#382—24no—
88#382—22no—
89#382—21no—
90#382—21no—
91#382—22no—
92#382—24no—
93#382—42no—
94#382D412yes—
95#380—24no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D412yes—
99C4×C4⋊C4D412yes—
100C4×C22⋊C4—22no—
101#376—21no—
102#376—22no—
103#376Q814yes—
104C2×#95—22no—
105C2×#95—24no—
106C2×#95—22no—
107C43—21no—
108#343—42no—
109#343—42no—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—22no—
114#339—24no—
115#339—24no—
116#339—24no—
117#339—24no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D414yes—
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C2×M16—22no—
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16C2×Q814yes—
136C2×M16C2×Q812yes—
137C2×M16C2×Q812yes—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141#125—88no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4—88no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×Q814yes—
192C2×C42C2×Q812yes—
193C2×C42C2×Q812yes—
194C2×C42C2×Q812yes—
195C2×C42#4114yes—
196C2×C42#4114yes—
197C2×C42#4112yes—
198#107—42no—
199#107—42no—
200#106—42no—
201#106—42no—
202C4⋊C8—28no—
203#97—24no—
204C4×C8—24no—
205#95—28no—
206#95—28no—
207#95—22no—
208#95—22no—
209#95—24no—
210#95—28no—
211#95—48no—
212#95—42no—
213#95—42no—
214C24C22×D411yescharacteristic
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—48no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4#12012yes—
237C22×C4#12012yes—
238C22×C4C22⋊Q814yes—
239C22×C4C4⋊Q814yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×Q811yescharacteristic
243C22×C4C22×Q811yescharacteristic
244C22×C4C22×Q811yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4112yes—
247C22×C4C2×#4112yes—
248C22×C4H(5,𝔽2)12yes—
249C22×C42−1+412yes—
250M16—24no—
251M16—24no—
252M16—24no—
253M16—24no—
254M16—42no—
255M16—42no—
256M16—42no—
257M16—42no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—44no—
276C4⋊C4—48no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—84no—
289C4⋊C4—88no—
290C4⋊C4—88no—
291C4⋊C4—816no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—44no—
300C42—44no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42C22⋊Q812yes—
306C42C22⋊Q812yes—
307C42C22⋊Q814yes—
308C42#12518yescomplemented
309C42C4⋊Q814yes—
310C42C4⋊Q812yes—
311C42C4⋊Q812yes—
312C23—22no—
313C23—22no—
314C23C2×#12011yescharacteristic
315C23C2×C22⋊Q811yescharacteristic
316C23C2×C4⋊Q811yescharacteristic
317C23#54612yes—
318C23#54812yes—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4#39514yes—
351C2×C4#39514yes—
352C2×C4C2×C22⋊Q811yescharacteristic
353C2×C4C2×C22⋊Q811yescharacteristic
354C2×C4C2×#12512yes—
355C2×C4C2×C4⋊Q811yescharacteristic
356C2×C4C2×C4⋊Q811yescharacteristic
357C2×C4#53211yescharacteristic
358C2×C4#53211yescharacteristic
359C2×C4#53312yes—
360C2×C4#53511yescharacteristic
361C2×C4#53611yescharacteristic
362C2×C4#54412yes—
363C2×C4#54412yes—
364C2×C4#54712yes—
365C2×C4#55012yes—
366C2×C4#55212yes—
367C2×C4#55312yes—
368C2×C4#55412yes—
369C2×C4#55712yes—
370C8—42no—
371C8—42no—
372C8—44no—
373C22—22no—
374C22—22no—
375C22—24no—
376C22—22no—
377C22—22no—
378C22C2×#39511yescharacteristic
379C22#243111yescharacteristic
380C22#243211yescharacteristic
381C22#244212yes—
382C22#255012yes—
383C22#301912yes—
384C22#302012yes—
385C4—24no—
386C4—24no—
387C4—48no—
388C4—82no—
389C4—84no—
390C4—81no—
391C4—84no—
392C4—81no—
393C4—161no—
394C4—161no—
395C4#244112yes—
396C4#255212yes—
397C4#256511yescharacteristic
398C4#256611yescharacteristic
399C4#266011yescharacteristic
400C4#285911yescharacteristic
401C2—22no—
402C2—22no—
403C2#1522811yescharacteristic
404C2#2356112yes—
405C2#2559311yescharacteristic
406C2#2559411yescharacteristic
407C2#2566912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
308. C42#125semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352862
#211722
#207652
#143542
#113402
#113392
#112112
#112102
#108572
#108402
#107012